#152860

order	512
nilpotency class	5
isoclinism	isoclinic class of #151127
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2177
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#509
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

180 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2)(3,4)(6,11,14,25)(7,16)(8,20)(9,21)(12,23,29,27)(13,31)(15,33,34,35)(17,28)(18,22)(19,32)(24,36,30,26);(2,4)(6,12,14,29)(7,17)(11,23,25,27)(13,32)(15,30,34,24)(16,28)(19,31)(26,35,36,33);(5,6,13,33,8,15,7,11,21,29,16,26,18,24,31,23,10,14,32,35,22,34,17,25,9,12,28,36,20,30,19,27)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=egh,e^2=ghi,g^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=ch,e^b=ei,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319455, #72319598, #72319601
maximal normal subgroup of	#72319455, #72319598, #72319601

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60056);